What is the Range of $w(w+x)(w+y)(w+z)$ When $x+y+z+w=x^7+y^7+z^7+w^7=0$?

[anemone]

Here is this week's POTW:

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Determine the range of $w(w+x)(w+y)(w+z)$ where $x,\,y,\,z$ and $w$ are real numbers such that $x+y+z+w=x^7+y^7+z^7+w^7=0$.

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[anemone]

Congratulations to lfdahl for his correct solution, which you can find below:

Spoiler

The range is zero.

Using the two conditions: $w= -(x+y+z)$ implies $w^7= -(x+y+z)^7 = -x^7-y^7-z^7$.

One possible solution is of course letting all four variables be zero: $w = x = y = z = 0$.

Another is letting two variables´ sum be zero: Either

$ x + y= 0 \Rightarrow w = -z $ or

$ x + z = 0 \Rightarrow w = -y $ or

$ y + z = 0 \Rightarrow w = -x$. In any of the cases we get: $w(w+x)(w+y)(w+z) =0$.

The interval length of one point is zero.